LRST Interactive Dynamics Explorer

Core Parameters

L_symSymmetric Loss
L_sym = L_leak + L_diff + L_relax

Baseline dissipation before asymmetry is applied. The "resting" loss rate of the ionic lattice site.1.00

Drive GInput Drive
G ≈ G_Na + G_Ca_assist

Ionic input per cycle. In the canonical regime G=1.05 sits just above L_sym=1.00 — but this surplus does NOT guarantee growth when asymmetry acts.1.05

Threshold X_cCritical Threshold
State value where ε switches sign.
Below: ε=+ε₊ → L_eff rises → decay
Above: ε=−ε₋ → L_eff falls → growth

In hysteresis model, this splits into X_on and X_off.5.0

ε₊ (adds loss)Dissipative Asymmetry
L_eff = L_sym × (1 + ε₊)

At ε₊=0.10: L_eff=1.10, ΔX=−0.05/cycle

Slow bleed below threshold.+0.10

ε₋ (reduces loss)Retentive Asymmetry
L_eff = L_sym × (1 − ε₋)

At ε₋=0.10: L_eff=0.90, ΔX=+0.15/cycle

Key LRST prediction: ε₋ drives 3× stronger response than ε₊ penalises. Retention-pathway interventions dominate.−0.10

Cycles NIteration Count
Divergence grows linearly with N. A 10% asymmetry after 80 cycles produces ~16 unit gap from identical starting conditions.80

Initial Conditions

X₀ subSubthreshold Start
Starts below X_c. ε₊ applies immediately.
State decays despite G > L_sym.

The key demonstration: positive net drive is not sufficient when asymmetry raises L_eff above G.2.0

X₀ nearNear-threshold Start
Most sensitive condition — small changes flip the outcome. Reveals sharpness of the decision boundary.4.5

X₀ supraSuprathreshold Start
Starts above X_c. ε₋ applies immediately.
State grows persistently.

Same G and L_sym as sub-threshold — opposite outcome. Loss pathway structure determines destiny.6.5

Legend

Subthresholddecays

Near-thresholdmarginal

Suprathresholdgrows

Final State (N)Final State Readout
ΔX gap = X_supra − X_sub = macroscopic divergence from asymmetry alone. Canonical value ≈ 16 after 80 cycles.

X_sub—

X_near—

X_supra—

ΔX gap—

Export Results

"Crossing X_c changes the structure of loss — not just the magnitude of input."

X = ionic/excitatory state · L = leakage+diffusion+relaxation · ε₊ = adds loss (dissipative) · ε₋ = reduces loss (retentive) · G = input drive

State Trajectory · X(n) vs cycle nMain Trajectory Plot
Three starting conditions diverge at threshold. Gap grows linearly — iterative amplification of small asymmetry into macroscopic structure.

L_eff(X) — effective lossEffective Loss Curve
Red: below X_c, L_eff > L_sym
Green: above X_c, L_eff < L_sym
Blue dashed = G. Growth only where L_eff < G.

ΔX/cycle — net gainNet Gain per Cycle
ΔX = G − L_eff(X)
Red = decay, Green = growth.
Note asymmetry: retentive gain (+0.15) > dissipative loss magnitude (−0.05).

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Core Parameters

L_symSymmetric Loss
Both trajectories share identical L_sym — isolating the effect of pathway asymmetry from raw loss magnitude.1.00

Drive GInput Drive
G is identical for both trajectories after the pulse ends. Memory effect is demonstrated by history alone, not ongoing drive differences.1.05

ε₊ (adds loss)Dissipative ε₊
Active when below X_on or after falling below X_off.
L_eff = L_sym × (1 + ε₊)
No-pulse trajectory stays here permanently.+0.10

ε₋ (reduces loss)Retentive ε₋
Active after crossing X_on, until falling below X_off.
L_eff = L_sym × (1 − ε₋)
Self-reinforcing: lower L_eff keeps state high, keeps ε₋ active.−0.10

Hysteresis ThresholdsThe Hysteresis Pair
X_on: must exceed to enter retentive
X_off: must fall below to exit retentive

Gap = X_on − X_off = memory depth. Same current X can represent two different modes depending on history.

X_on — enter retentiveActivation Threshold
Without pulse: starts at X₀=4.0, never reaches X_on=5.0 → decays. With pulse: briefly pushed above X_on → mode flips → persists.5.0

X_off — exit retentiveDeactivation Threshold
X_off < X_on creates the gap zone. A system in retentive mode stays there even as X falls into the gap — path-dependence in action.4.2

Pulse ParametersTransient Stimulus
Brief extra drive simulating external stimulation. Key question: does the elevated mode persist after the pulse ends? If yes — LRST memory demonstrated.

X₀ (both)4.0

Pulse startPulse Start Cycle
Try increasing until pulse can no longer push X above X_on — defines the rescue window.8

Pulse duration4

Pulse amplitudePulse Amplitude
Reduce until pulse barely fails to cross X_on — this is the minimum effective stimulus. Below it, no memory forms regardless of duration.0.70

Cycles N60

Legend

No pulse — decaysbaseline

With pulse — persistsmemory

Pulse windowtransient

Memory Result

No-pulse Xf—

Pulse Xf—

X_on—

Gap Xon−XoffHysteresis Gap = Memory Depth
Larger gap = more robust memory = harder to erase. Systems in the gap zone stay retentive until pushed below X_off.—

Export Results

"A transient input can permanently alter system behavior — outcome depends on history, not only present state."

X = ionic state · X_on = enter retentive · X_off = exit retentive · gap = X_on − X_off = memory depth

Hysteresis Trajectory · identical starts, different historiesHysteresis Trajectory
Same X₀, same G — different outcomes based solely on history. After the pulse ends, identical ongoing drive produces opposite behavior.

Mode state over timeMode Timeline
Red = dissipative (ε₊), Green = retentive (ε₋). The mode flip in the pulse trajectory is irreversible under the same G.

Hysteresis loop · X vs L_effHysteresis Loop
The loop shape — different paths entering vs exiting high state — is the geometric signature of memory. Same X, two different L_eff values depending on history.

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"System behavior depends jointly on input drive and loss asymmetry — neither alone is sufficient."

Each panel sweeps one parameter across its full range, all others fixed at canonical values (G=1.05, L=1.00, X_c=5, ε±=0.10, N=80). Vertical axis = X_final after N cycles. Red=X₀=2 (sub), Amber=X₀=5 (near), Green=X₀=7 (supra). Dashed vertical = current slider value.

Sweep: ε₊ε₊ Sensitivity
Penalises sub-threshold only. Supra-threshold trajectory largely unaffected — ε₊ never activates above X_c.0.10

Range 0.01–0.5

Sweep: ε₋ε₋ Sensitivity
Strongly amplifies supra-threshold growth. Sub-threshold unaffected. Key LRST prediction: ε₋ dominates ε₊ in per-cycle effect.0.10

Range 0.01–0.5

Sweep: Drive GG Sensitivity
Two critical G values define the operating window: G_crit_ret (below which retentive fails) and G_crit_diss (above which sub-threshold grows).1.05

G range 0.5–2.5

Sweep: X_cBifurcation
When X_c sweeps past X₀=7, the supra-threshold trajectory abruptly flips from growth to decay. This is the LRST bifurcation point.5.0

Xc range 1–20 · bifurcation

Sweep: L_symLoss Ceiling
L_sym_max = G / (1 − ε₋) ≈ 1.17 at canonical values. Above this ceiling, no asymmetry can sustain growth.1.00

L_sym range 0.3–2.5

Sweep: Hyst. gap ΔX_hMemory Depth
Gap=0: single threshold. Larger gap: deeper, more robust memory. Near-threshold trajectory most sensitive.0.80

Gap = Xon − Xoff

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"These three experiments constitute the minimal publishable demonstration of LRST applied to ion-coupled substrate dynamics."

EXPERIMENT 1

Threshold Crossing & Bifurcation

Hypothesis: varying X₀ across X_c produces a sharp bifurcation in final state — demonstrating that loss pathway structure, not input magnitude, determines outcome.

Fixed: G=1.05, L_sym=1.00, ε+=0.10, ε−=0.10, N=80
Sweep: X₀ from 0 to 12 in steps of 0.5
Outcome: X_final vs X₀ — bifurcation at X_c=5

X_final vs initial condition X₀

Press Run to execute.

EXPERIMENT 2

Minimum Effective Stimulus

Hypothesis: there exists a critical pulse amplitude below which no memory forms regardless of duration — the minimum effective stimulus for LRST state transition.

Fixed: G=1.05, L=1.00, X₀=4.0, X_on=5.0, X_off=4.2, N=60
Sweep: pulse amplitude 0.05 to 1.50
Outcome: final state vs amplitude — critical threshold visible as step

X_final (pulse) vs pulse amplitude

Press Run to execute.

EXPERIMENT 3

Retention Asymmetry Dominance

Hypothesis: ε₋ (retentive asymmetry) produces disproportionately greater final-state divergence than ε₊ (dissipative asymmetry) — retention pathways dominate system destiny.

Fixed: G=1.05, L=1.00, X_c=5, X₀_sub=2, X₀_supra=7, N=80
Sweep: both ε+ and ε− independently from 0 to 0.5
Outcome: ΔX gap vs ε — ε− curve lies above ε+ curve

ΔX divergence vs asymmetry parameter

Press Run to execute.

"Structure of loss pathways determines system destiny more than input magnitude — a principle that applies across scales and domains."

What LRST Claims

Loss-Recovery Systems Theory formalises a general principle: small asymmetries in competing loss pathways can accumulate over repeated interactions to produce large-scale, macroscopic divergence.

The core equation is: X_n+1 = X_n + G − L_sym(1 + ε(X_n))

Where ε(X) is a state-dependent asymmetry parameter. Below threshold, ε is positive (adds loss). Above threshold, ε is negative (reduces loss). This single structural change in the loss term — not the drive — determines whether a state persists or decays.

What This Tool Can Determine

This visualiser provides a controllable, predictive model of asymmetric loss dynamics. It can determine:

• Whether a system will grow or decay from a given initial condition
• Whether a stimulus will persist or vanish after removal
• The minimum intervention required to switch a system's mode
• The stability of a retained state under perturbation
• The limits of recoverability — when a state is unrescuable
• The operating window of drive G for bistable behaviour

Biological Substrate Mapping

Applied here to ion-coupled protein lattices:

X = local ionic excitatory state of a tubulin lattice site
G = effective Na⁺/Ca²⁺ ionic drive per cycle
L_sym = passive leakage + diffusion + relaxation
ε₊ = sub-threshold channel bias toward leakage
ε₋ = supra-threshold Ca²⁺-dependent retention
X_on/X_off = activation/deactivation thresholds for gating commitment

The 0.5 mM Ca²⁺ threshold observed in 1997 ion dynamics data maps directly to X_c in this framework.

Domains of Application

The asymmetric loss principle is domain-agnostic. Any system with competing gain and loss pathways where the loss structure is state-dependent will exhibit LRST-class dynamics:

Neural Systems

Energy Devices

Biomaterials

Economics

Spacecraft Systems

Ecological Models

Ion Channel Gating

Quantum Substrates

Scope and Limits

LRST does:
• Provide a quantitative tool for asymmetry-driven systems
• Predict scaling behaviour under iteration
• Identify critical thresholds for growth vs decay
• Describe hysteretic memory without external storage

LRST does not:
• Derive fundamental physical constants
• Replace domain-specific mechanistic theories
• Claim specific microtubule memory mechanisms without validation
• Explain particle physics asymmetries mechanistically

Citation & Framework

Framework: Loss-Recovery Systems Theory (LRST)
Extended model: Unified MultiChannel Substrate Framework (UMCSF)
Recovery equation: R = ρ · L · σ · α
Author: James Ball, Ottawa, Ontario
Tool version: LRST Interactive Dynamics Explorer v2.4

This tool was developed as an in silico validation instrument for the biological substrate application of LRST, using retrospective validation against published ion dynamics data.